Identifying the dynamics of complex
spatio-temporal systems by spatial
recurrence properties
Chiara Mocenni
Department of Information Engineering
Centre for the Study of Complex Systems
University of Siena
[email protected]
in collaboration work with A. Facchini and A.Vicino
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Outline of the talk
Complex spatio-temporal dynamical systems;
State space reconstruction from time series and
spatio-temporal time series;
Recurrence plots: definition and measures;
DET − ENT diagram for the classification of complex
2D spatio-temporal systems;
Structural changes in time and space dynamics;
Application to the Complex Ginzburg-Landau
Equation;
Application to the Schnackenberg reaction-diffusion
system;
Conclusions and future research.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Spatio-temporal complex systems
Spatially extended systems may exhibit irregular
behavior both in space and time leading to
spontaneous emergence of spatial patterns: Turing
structures, traveling and spiral waves, turbulence.
Reaction-diffusion equations have been used for
describing the main physical mechanisms leading to
such phenomena.
One main and still investigated problem is dealing
with a partially unknown system of which only the
observations of some of its spatial variables are
available.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
State-space reconstruction
Reconstructing the state space of a dynamical
system consists with identifying its dynamics using a
set of measurements.
The starting point of the embedding theorem1 for time
series is that in nonlinear systems every observed
variable include, in an unknown way, the information
of all the others.
The concept of recurrence is strictly related to that of
dynamical systems, as originally stated by Poincaré.
1
Takens F., “Detecting strange attractors in turbulence”, Lecture
Notes in Math. Springer New York (1981).
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The case of time series
Given a time series [s1 , . . . , sN ], where si = s(i∆t)
and ∆t is the sampling time, the system dynamics
can be reconstructed using the theorem of Takens
and Mañe.
The reconstructed trajectory X is expressed as a
matrix in which each row is a phase space vector
xi = [si , si+τ , . . . , si+(DE −1)τ ],
i = 1, . . . , N − (DE − 1)τ , where DE is the embedding
dimension.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
1D Recurrence Plot
The Recurrence Plot (RP), proposed for the first time
by Eckmann et al. (1987), is a visual tool able to
identify temporal recurrences in multidimensional
phase spaces.
In the RP, any recurrence of state i with state j is
pictured on a boolean matrix expressed by:
RDi,jE = Θ( − ||xi − xj ||) ,
(1)
where xi,j ∈ RDE are embedded vectors, i, j ∈ N, Θ(·)
is the Heaviside step function and is an arbitrary
threshold.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Examples of Recurrence Plot
1000
3000
900
2900
800
2800
2700
600
time (in samples)
time (in samples)
700
500
400
2600
2500
2400
300
2300
200
2200
100
2100
0
0
200
400
600
time (in samples)
800
1000
2000
2000
2100
2200
2300
2400
2500
2600
time (in samples)
2700
2800
Figure: Recurrence Plots of periodic, random and chaotic
signals.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
2900
3000
Spatio-temporal time series
Analogously to time series, it can be assumed that
the evolution of a certain region of a spatially
distributed complex system depends in some way by
all the other regions.
The problem of understanding the dynamics of
spatio-temporal dynamical system may be
investigated by identifying the spatial state
recurrences in the spatial domain.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Spatial Recurrence Plots
Given a d dimensional cartesian system, the
n-dimensional RP2 , 3 is
R~ı,~ = Θ( − ||~x~ı − ~x~||)
where ~ı = i1 , i2 , . . . , id is the d-dimensional coordinate
vector and ~x~ı is the associated phase-space vector.
The Line of Identity is given by R~ı,~ = 1, ∀~ı = ~, and is
represented by an hypersurface.
2
N. Marwan, J. Kurths and P. Saparin, “Generalised recurrence
plots analysis for spatial data”, Phys. Lett. A, 360, pp. 545-551 (2007)
3
D. B. Vasconcelos, S. R. Lopes, R. L. Viana and J. Kurths,
”Spatial recurrence plots“, Physical Review E, 73, pp. 1-10 (2006)
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Application to 2D systems
The discretized solutions of a 2D spatio-temporal system
at fixed time can be represented by two-dimensional
cartesian objects (images) composed of scalar values,
therefore the GRP
Ri1 ,i2 ,j1 ,j2 = Θ( − ||xi1 ,i2 − xj1 ,j2 ||)
defines a four dimensional RP containing a
two-dimensional LOI plane, where xi1 ,i2 identifies a pixel of
the image.
Two states are recurrent if the associated pixels xi1 ,i2 and
xj1 ,j2 are within the threshold .
The line structures become 2-dimensional.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Recurrence Rate (RR)
Analogously to the one dimensional case, we define the
Generalized Recurrence Quantification Analysis (GRQA)
measures based on the histogram P(l) of the line lengths:
N
N
1 X
1 X
RR = 4
Ri1 ,i2 ,j1 ,j2 = 4
lP(l).
N
N
i1 ,i2 ,j1 ,j2
l=1
RR is the fraction of recurrent points with respect to the
total number of possible recurrences. It is a density
measure of the RP.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Determinism (DET )
PN
lP(l)
l=l
DET = PN min
,
l=1 lP(l)
where lmin is the minimum length considered for the
diagonal structures.
DET is the fraction of recurrent points forming diagonal
structures with respect to all the recurrences4 .
4
In the 1D framework, a line of length l indicates that, for l time
steps, the trajectory in the phase space has visited the same region at
different times.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Entropy (ENT )
ENT = −
N
X
p(l) log p(l),
l=lmin
P(l)
p(l) = PN
.
l=lmin P(l)
ENT is a measure of the distribution of the diagonal lines
in the GRP.
It refers to the Shannon entropy with respect to the
probability to find a diagonal line of exactly length l 5 .
5
For periodic signal or uncorrelated noise the value is small
(∼ 0.2 − 0.8), while for chaotic systems, e.g. Lorenz, ENT ∼ 3 − 4.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The spatial recurrence properties
We have proposed to use DET and ENT for the
analysis of spatially distributed dynamical systems by
looking at the spatial recurrence properties of the
system, and, in particular, by seing the available
snapshots as solutions of an unknown 2D system at
a fixed regime time.
The idea is that some signatures of the system may
be identified by evaluating the spatial properties of
the solution at a fixed time.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Examples: fractals and chemotaxis
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Examples: Turing structures in the BZ
reaction
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Examples: chlorophyll distribution in oceans
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Examples: periodic patterns
,
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Histograms of the line lengths distribution
(a)
(b)
25
25
Turing Patterns
20
20
15
15
log(Nl)
log(Nl)
Uniform Noise
Linear fit
10
5
0
10
5
4
5
6
7 8 9 10 11 12 13
Line length
0
0
20
40
Line length
60
80
(a) White noise: the line lengths are exponentially
distributed and the maximum length is short;
(b) Turing patterns: In the beginning an exponential
distribution is found, while in the remaining part the
histogram is more complex.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
ENT and DET indicators for the classification
of complex images
DET is a measure of the global appearance of the image:
values of determinism larger than 60-70% indicate that
the image has strong recurrent components;
ENT accounts for the local organization: periodic
distribution of the diagonals shows low ENT values, since
the distribution is trivial; a random distribution of the
diagonal structures produces a low entropy value;
We introduced the the DET − ENT diagram to
characterize the images according to their recurrence
properties.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The DET − ENT diagram
(a)
3
D
2.5
F
2
ENT
E
1.5
C
A
1
0.5
0
B
10
20
30
40
DET
50
60
70
80
(b)
3
Turing
Fractals (small)
Fractals (big)
Chlorophyll
Diffusion waves
Random
Periodic
Dict. Discoideum
D
2.5
F
2
ENT
E
1.5
C
A
1
0.5
0
B
10
20
30
Chiara Mocenni
40
DET
50
60
70
80
Automatica.it, September 7-9, 2011, Pisa
Detecting changes in the dynamics
Is it now possible to analyze and detect structural
changes in the spatio-temporal dynamics of a partially
unknown system using a limited number of information on
temporal evolution of its spatial variable?
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Complex Patterns in spatial systems
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The Complex Ginzburg-Landau Equation
(CGLE)
We use GRQA and the DET − ENT diagram for
investigating the dynamics of the Complex
Ginzburg-Landau Equation.
The Complex Ginzburg-Landau Equation displays a
rich spectrum of dynamical behaviors describing a
large variety of physical systems, such as nonlinear
waves, second order phase transitions,
superconductivity, superfluidity, Bose-Einstein
condensation and liquid crystals.
It is a prototypical example of pattern formation (see
previous slide) and presents bifurcations.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The CGLE equation
The CGLE reads:
∂t A = A+(1+ıa)∇2 A−(b−ı)|A|2 A
a, b ∈ R
(2)
The first term of the rhs is related to the linear instability
mechanism leading to oscillation. The second term
accounts for diffusion and dispersion, while the cubic term
insures, for b > 0, the saturation of the linear instability
and is involved in the renormalization of the oscillation
frequency.
In two dimensions, the solutions of the CGLE are families
of plane waves. Their behavior in the parameter space
(a, b) is very complex and still under investigation.
Chiara Mocenni
A(x, y) ∈ C,
Automatica.it, September 7-9, 2011, Pisa
The bifurcation curve in parameter space
2.5
2
1.5
a
1
Unstable spirals
0.5
e
zon
ion
nsit
Tra
0
−0.5
Stable spirals
−1
−1.5
0
0.2
0.4
0.6
0.8
b
1
1.2
1.4
1.6
Behavior in the parameter space of the real part of A.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The DET − ENT diagram (1/2)
(a)
2.5
2
1.5
1
S1
a
0.5
0
−0.5
−1
−1.5
0
0.5
1
1.5
b
(b)
4.5
4
ENT
3.5
3
2.5
S2
2
1.5
0
10
20
30
40
50
DET
Distribution of 80 points in plane (b, a) (a); Clustering (b).
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The DET − ENT diagram (2/2)
The three regions are clearly identifiable in the
DET − ENT diagram, where the clusters of stable
and unstable spirals are clearly separated by an
intermediate region corresponding to the transition
zone above the curve S1 .
The curve S1 itself corresponds to the curve S2 in the
DET − ENT diagram and the cluster of the transition
zone lays on this curve.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
New experiments and method refinement
The CGLE was integrated in a square domain of
L = 512 points with periodic boundary conditions.
A portion of the phase plane ranging from
a = [−1.5, 1.5] and b = [−1.5, 1.5] is considered.
Starting from random initial conditions, the whole
trajectory of the system is initially analyzed for each
value of a and b.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Temporal evolution of DET and ENT
30
25
D
20
15
10
α=−1, β=−1
α=−1, β=0.1
α=−1, β=1
5
0
0
100
200
1000
Iterations
2000
3000
4000
3.5
3
E
2.5
2
1.5
1
α=−1, β=−1
α=−1, β=0.1
α=−1, β=1
0.5
0
0
100
200
Chiara Mocenni
1000
Iterations
2000
3000
4000
Automatica.it, September 7-9, 2011, Pisa
A sensitivity function
"
K (b) =
∆ENT
∆b
2
Chiara Mocenni
+
∆DET
∆b
2 #1/2
.
Automatica.it, September 7-9, 2011, Pisa
Clustering
4
B
3.5
E
3
2.5
G
A
2
1.5
5
10
15
20
25
30
35
40
D
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The DET − ENT diagram
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Bifurcations detection (1/3)
In the DET − ENT diagram the zones A and B are
separated by a transition zone.
The lines bounding the regions A correspond, with
very good agreement, to the line S1 : the boundary of
the convective instability of the spiral waves, also
known as the Eckhaus limit.
The transition region G separating clusters B and A is
found to separate the regions A and B in the (a, b)
plane.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Bifurcations detection (2/3)
1.5
1
A
G
Line S1, as in [16]
a
0.5
B
0
−0.5
G
A
−1
−1.5
−1.5
−1
−0.5
Chiara Mocenni
0
b
0.5
1
Automatica.it, September 7-9, 2011, Pisa
1.5
Bifurcations detection (3/3)
1.5
4
B
1
A
3.5
G
Line S1, as in [16]
0.5
E
a
3
2.5
B
0
G
−0.5
G
A
A
2
−1
1.5
5
10
15
20
25
30
35
40
D
−1.5
−1.5
−1
−0.5
0
b
0.5
1
1.5
A cluster jump in the DET − ENT diagram corresponds to
crossing a bifurcation line in the parameter space.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
The Schnakenberg system
Describes a simple chemical reaction showing limit cycle
behavior and Turing instabilities. The equations reads:
∂t u = γ(k1 − u + u 2 v ) + ∇2 u,
∂t v = γ(k2 − u 2 v ) + d∇2 v ,
u(x, y, t), v (x, y , t) ∈ R;
x, y are the spatial variables;
γ is proportional to the spatial domain size;
k1 and k2 depend on the reaction rates;
d is the ratio of the diffusions of the two reactants.
The critical diffusion coefficient dc depends on k1 and k2 .
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Detecting Turing bifurcations in the
Schnakenberg system (1/2)
(a)
dc
50
D*
0
9.5
5
10
10.5
11
11.5
12
dc
12.5
d
(b)
13
13.5
14
14.5
15
4
E
D
100
3
2
1
9.5
E*
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
d
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Detecting Turing bifurcations in the
Schnakenberg system (2/2)
60
k1=0.30
D*(k )
55
1
quadratic fit
50
k1=0.15
1
D*(k )
45
40
k1=0.50
35
30
25
20
0.1
0.2
0.3
Chiara Mocenni
k1
0.4
0.5
0.6
Automatica.it, September 7-9, 2011, Pisa
Conclusions
We proposed the DET − ENT diagram for the
analysis of complex patterns;
The method identifies the essential characteristics,
including structural changes, of a complex
spatio-temporal dynamical system by analyzing
instantaneous spatial measurements at steady state;
The application of the GRQA to the solutions of the
CGLE and to the Schnackenberg system led to the
identification of bifurcation lines in the parameter
space.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
Future Research
Solving inverse problems for the reconstruction of
ocean plankton dynamics and turbulent patterns from
remote sensing images.
Identification of the dynamics in the field of systems
biology, such as spatial modeling of tumor growth and
cell diseases, brain cancer, where the spatial data
are provided by biopsy and Functional Magnetic
Resonance imaging (FMRi) techniques.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa
For Further Reading
C. Mocenni, A. Facchini,A. Vicino, “Identifying the
dynamics of complex spatio-temporal systems by
spatial recurrence properties” Proc. Nat. Academy of
Sciences, 107, 8097-8102, 2010.
A. Facchini, F. Rossi, and C. Mocenni, “Spatial
recurrence strategies reveal different routes to Turing
pattern formation in chemical systems”, Phys. Lett.
A, 373:4266-4272, 2009.
A. Facchini, C. Mocenni and A. Vicino, “Generalized
Recurrence Plots for the analysis of images from
spatially distributed systems”, Physica D, vol. 238,
pp. 162-169, 2008.
Chiara Mocenni
Automatica.it, September 7-9, 2011, Pisa